李型有限群的非分歧上同调

Rost nilpotence and higher unramified cohomology

We develop an approach to proving the Rost nilpotence principle involving higher unramified cohomology. We use this to prove the principle for certain varieties of dimension $\leq 3$ over a perfect field.

一般对角超曲面的低次非分歧上同调

Low degree unramified cohomology of generic diagonal hypersurfaces

本文证明在维数大于 i 的射影空间中，一般对角超曲面的第 1、2、3 次非分歧上同调群均为平凡群。

Motivic classes of classifying stacks of finite groups and unramified cohomology

Combining work of Peyre, Colliot-Thélène and Voisin, we give the first example of a finite group $G$ such that the motivic class of its classifying stack $BG$ in Ekedahl's Grothendieck ring of stacks over $\mathbb{C}$ is non-trivial and $BG$ has trivial unramified Brauer group.

截断前推与精细化的非分歧上同调

Truncated pushforwards and refined unramified cohomology

本文证明了精细化的非分歧上同调典范同构于Zariski层自然截断复形的超上同调，推广了Bloch-Ogus的经典结果并解决了Kok和Zhou的猜想。

The derived functors of unramified cohomology

We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.

On the mod $p$ unramified cohomology of varieties having universally trivial Chow group of zero-cycles

Auel-Bigazzi-Böhning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as well. In this paper, we generalize their argument to arbitrary unramified mod $p$ étale motivic cohomology groups. We also see that the properness assumption on the variety $X$ can be dropped off by using the Suslin homology together with a certain tame subgroup of the unramified cohomology group.

例外simply connected群分类簇的非分歧上同调

Unramified cohomology of classifying varieties for exceptional simply connected groups

设BG是例外单连通代数群G的分类簇.本文计算了任意域F上BG的3阶非分歧Galois上同调，其值为Q/Z（2）。结合Merkurjev的一篇论文，完成了全域上单连通G半单的这些上同调群的计算.这些计算提供了一个简单单连通群G的另一个例子，使得BG不是稳定有理数。

关于非闭域上非分歧上同调的计算

On computation of unramified cohomology over non-closed fields

本文给出在非代数闭域上具有非平凡非分歧上同调的簇的例子，包括有界上同调维数域上的簇以及特征非2的任意域上有理曲面上的圆锥丛。

三次非分歧上同调群

Degree Three Unramified Cohomology Groups

设$p$为奇素数。Peyre证明了存在一个阶为$p^{12}$的群$G$，使得$H_{nr}^3(\bm{C}(G), \bm{Q}/\bm{Z})$是非平凡的。利用Peyre方法，我们能够证明对于某些$p^9$阶群，同样的结论也成立.

On the unramified cohomology of certain quotient varieties

In this note, we consider unramified cohomology with $\mathbb{Z}/2$ coefficients for some (degree two) quotient varieties and describe a method that allows one to prove the non-vanishing of these groups under certain conditions. We apply this method to prove a non-vanishing statement in the case of Kummer varieties. Combining this with work of Colliot-Thélène and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture.

Unramified cohomology, integral coniveau filtration and Griffiths group

We prove that the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first piece is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 subgroup, and whose next graded piece is controlled by the Griffiths group Griff^{k/2+1}(X) when k is even and is related to the higher Chow group CH^{(k+3)/2}(X, 1) when k is odd. The first piece is a generalization of the Artin-Mumford invariant (k=2) and the Colliot-Thelene-Voisin invariant (k=3). We also gi…

On some finiteness results for unramified cohomology

We obtain several finiteness results for the unramified cohomology of function fields of algebraic varieties defined over fields of type (F'_m), a class that includes algebraically closed fields, finite fields, local fields, and some higher local fields of characteristic 0.

Grothendieck环的非分歧上同调、$\A^1$ -连通性和Chevalley-Warning问题

Unramified cohomology, $\A^1$-connectedness, and Chevalley-Warning problem in Grothendieck ring

研究了Grothendieck环$K_0(Var/k)$上的Chevalley-Warning问题.我们表明，$\A^1$ -同伦理论产生良好定义的不变量上$K_0(Var/k)/\bold{L}$，特别是布劳尔群是这样一个不变量。我们用它来给$C_1$ -域上的Chevalley-Warning猜想一个具体的反例（Brown and Schnetz，2011）。这也对Bilgin（2011）的问题给出了否定的答案。

非分歧上同调与Brauer--Manin配对

Unramified cohomology and Brauer--Manin pairing

本文证明局部域上光滑射影曲面非分歧上同调的Tate模在适当条件下消失，并应用于Brauer--Manin配对的核问题，通过建立Bloch公式实现。

On Chow-pure cohomology and Euler characteristics for motives and varieties, and their relation to unramified cohomology and Brauer groups

We study Grothedieck groups of triangulated categories using weight structures, weight complexes, and the corresponding pure (co)homological functors. We prove some general statements on $K_0$ of weighted categories and apply it to Voevodsky motives endowed with so-called Chow weight structures. We obtain certain "motivic substitutes" for smooth compactifications of smooth varieties over arbitrary perfect fields; this enables us to make certain unramified cohomology and Euler characteristic calculations that are closely related to results of T. Ekedahl and B. Kahn.

三次非分歧上同调与Noether问题

Unramified cohomology of degree 3 and Noether's problem

设$G$是有限群，$W$是{ \bf C}上$G$的忠实表示.群$G$作用于有理函数$\mathbf C(W)$的域上。当$G$为奇数阶群时，利用$G$的上同调刻画了不变函数域$\mathbf C(W)^G$的3次非分歧上同调群.这使我们能够给出一个群的例子，对于这个群，这个域不是有理数的，尽管它的非分歧的布劳尔群是平凡的。

Varieties that are not stably rational, zero-cycles and unramified cohomology

This is a survey of recent examples of varieties that are not stably rational. We review the specialization method based on properties of the Chow group of zero-cycles used in these examples and explain the point of view of unramified cohomology for the construction of nontrivial stable invariants of the special fiber. In particular, we find an explicit formula for the Brauer group of fourfolds fibered in quadrics of dimension two over a rational surface.

Degree 3 unramified cohomology of classifying spaces for exceptional groups

Let $G$ be a reductive group defined over an algebraically closed field of characteristic $0$ such that the Dynkin diagram of $G$ is the disjoint union of diagrams of types $G_{2}, F_{4}, E_{6}, E_{7}, E_{8}$. We show that the degree $3$ unramified cohomology of the classifying space of $G$ is trivial. In particular, combined with articles by Merkurjev \cite{Mer17} and the author \cite{Baek}, this completes the computations of degree $3$ unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type.

特征2下二次曲面的非分歧上同调

Unramified Cohomology of Quadrics in Characteristic Two

本文研究特征2域上光滑射影二次曲面的2、3次2-准素挠非分歧上同调，完全确定基域上同调到其非分歧上同调自然映射的核与余核，推广了特征非2情形的经典结果。

Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields

We formulate and analyze several finiteness conjectures for linear algebraic groups over higher-dimensional fields. In fact, we prove all of these conjectures for algebraic tori as well as in some other situations. This work relies in an essential way on several finiteness results for unramified cohomology.

Unramified logarithmic Hodge-Witt cohomology and $\mathbb{P}^1$-invariance

Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-Rülling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.

局部上同调消失于非分歧混合特征

Vanishing of local cohomology in unramified mixed characteristic

本文在非分歧混合特征环境下扩展了 Faltings 关于理想上同调维数的上界结果，并证明该上界是紧致的。

$A_{\ell}$型有限单群的非分歧Brauer群

Unramified Brauer groups of finite simple groups of Lie type $A_{\ell}$

本文研究了H^2（G，Q/Z）的子群B_0（G），它由对G的任意Abel子群有平凡限制的所有元素组成.群B_0（G）是代数簇V/G稳定合理性的最简单的非平凡障碍，其中V是群G的一个忠实复线性表示。本文证明了对于有限的李型A_{\ell}单群，B_0（G）是平凡的.

曲面的torsion birational motives与unramified cohomology

Torsion birational motives of surfaces and unramified cohomology

该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。

Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature $(1,n-1)$

In [Inventiones mathematicae, 184 (2011)], Vollaard and Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport-Zink space with signature $(1,n-1)$. They constructed an isomorphism between the closure of a stratum, called a closed Bruhat-Tits stratum, and a Deligne-Lusztig variety which is not of classical type. In this paper, we describe the $\ell$-adic cohomology groups over $\overline{\mathbb Q_{\ell}}$ of these Deligne-Lusztig varieties, where $\ell \not = p$. The computations involve the spectral sequence associated to the Ekedahl-Oort stratification o…

The Cohomology of Unramified Rapoport-Zink Spaces of EL-type and Harris's Conjecture

We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, μ}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, μ}(ρ)$ for $ρ$ a s…

Refined unramified cohomology of schemes

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low codimensions on smooth projective varieties previous results of Bloch--Ogus, Colliot-Thélène--Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1. This establishes a torsion version of a conjecture of Janns…

Degree three unramified cohomology groups and Noether's problem for groups of order $243$

Let $k$ be a field and $G$ be a finite group acting on the rational function field $k(x_g : g\in G)$ by $k$-automorphisms defined as $h(x_g)=x_{hg}$ for any $g,h\in G$. We denote the fixed field $k(x_g : g\in G)^G$ by $k(G)$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is well-known that if $C(G)$ is stably rational over $C$, then all the unramified cohomology groups $H_[nr}^i(C(G),Q/Z)=0$ for $i \ge 2$. Hoshi, Kang and Kunyavskii [HKK] showed that, for a $p$-group of order $p^5$ ($p$: an odd prime number), $H_[nr}^2(C(G),Q/Z)\neq 0$ if and only if …

有限域上三次非分岐上同调的消没

Vanishing of degree $3$ unramified cohomology over finite fields

本文证明有限域上费马三次曲线三次非分岐上同调消没及循环映射满射，并推广至满足 Tate 猜想的簇，应用于零循环局部-全局原理。

有限Krull维数的优秀局部无分歧正则环的局部上同调模的相伴素数的有限性

Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension

本文证明了Huneke和Lyubeznik关于局部上同调模相伴素理想有限性的猜想，在有限Krull维数的优秀局部无分歧正则环等条件下取得了实质性新进展。

Cohomologie non ramifiée dans le produit avec une courbe elliptique

A method of Gabber (2002) produces unramified cohomology classes in the products of certain varieties with an elliptic curve. The connection between third unramified cohomology and integral Hodge conjecture for codimension 2 cycles (2012, which builds upon results from algebraic K-Theory, then gives many examples of such a product for which this conjecture fails. The special case of the product with an Enriques surface was established by Benoist and Ottem (2018).

利用上同调不变量分离实代数簇的分支

Separating components of real algebraic varieties by cohomological invariants

该研究针对实数域上的光滑代数簇，利用等变上同调、非分歧上同调及非分歧二次型符号分离其实点连通分支，将结果应用于多种代数簇并得到$\boldsymbol{I}$-上同调相关新界及最优性结论。

圈积的等倾性和稳定上同调

Isoclinism and stable cohomology of wreath products

利用Hall对有限p-群引入的同倾的概念，我们证明了许多重要的有限p-群类都具有可由交换子群检测到的稳定上同调，见定理4.4。证明了循环群ZZ/p的n重圈积的稳定上同调可由初等交换p-子群检测，并给出了相应的上同调代数的具体描述.给出了它在计算李型有限群的非分歧稳定上同调中的一些应用。

导出非分支上同调的一个余积障碍

A coproduct obstruction for derived unramified cohomology

本文证明在指数特征可逆后，第一个导出非分支函子与任意小直和交换，并描述第二个导出非分支函子的范畴障碍，应用于曲面的扭阶界和对应检测。

Unramified Brauer group and its applications

This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.

$\mathbb A^1$-不变平展层的驯顺与曲线驯顺上同调

Tame and curve-tame cohomology of $\mathbb A^1$-invariant étale sheaves

本文在附加假设下将非分歧曲线驯顺上同调推广至$\mathbb{A}^1$-不变平展层，定义其与苏斯林同调的配对，并证明正特征域上光滑几何连通簇的该上同调等于基域上同调。

Birational motives, II: Triangulated birational motives

We develop birational versions of Voevodsky's triangulated categories of motives over a field, and relate them with the pure birational motives studied in arXiv:0902.4902 [math.AG]. We also get an interpretation of unramified cohomology in this framework, leading to "higher derived functors of unramified cohomology".

On the generic part of the cohomology of local and global Shimura varieties

Using the work of Fargues-Scholze, we prove a vanishing theorem for the generic unramified part of the cohomology of local Shimura varieties of general linear groups. This gives an alternative approach to vanishing results of Caraiani-Scholze for the cohomology of unitary Shimura varieties.

函数域非分歧扩张上的Azumaya代数

Azumaya algebras over unramified extensions of function fields

本文利用Milnor-Quillen $K$-群解释函数域的étale上同调挠部分，证明了在范数映射的温和条件下，光滑簇上存在可由Azumaya代数表示的上同调Brauer类，从而部分回答了Grothendieck的问题。

积分Hodge/塔特猜想在与投影超曲面的乘积上的失效

On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

本文通过细化无 ramifications cohomology方法，证明了Enriques面与光滑奇维投影超曲面的乘积上积分Hodge/塔特猜想的失效，并推广了Shen的结果到一般代数闭域。

齐性空间的非分歧代数Brauer群

Groupe de Brauer non ramifié algébrique des espaces homogènes (The unramified algebraic Brauer group of homogeneous spaces)

利用线性代数群$G$的Galois上同调约化为某个有限子商的Galois上同调，我们给出了计算具有$G'$的半单连通齐性空间$V=G\backslash G'$的非分歧代数Brauer群的不同公式，包括有限域和任意特征域$0$ .

Torsion 1-cycles and the coniveau spectral sequence

We relate the torsion part of the Abel-Jacobi kernel in the Griffiths group of 1-cycles to a birational invariant analogous to the degree 4 unramified cohomology and an invariant associated to the generalized Hodge conjecture in degree 2dim(X)-3. We also describe in terms of H-cohomology the Griffiths group of 1-cycles and the group of torsion cycles algebraically equivalent to zero of arbitrary dimension.

p-adic域上全退化的algebraic varieties的余上同调

On the etale cohomology of algebraic varieties with totally degenerate reduction over p-adic fields

设K是Q_p的有限扩张，X是K上的光滑射影簇.定义了K的整数环上X的正则真模型的特殊纤维的全退化约化和相应的Chow复形。若X有这样的约化，则证明了对任意l，X的Q_l-进余同调有一个滤子，其分次积作为Galois模同构于有限维Q-向量空间（具有有限的Galois非分歧作用）的张量积，且Q_l的扭具有分圆特征标.

李-* 代数的半无限上同调

Semiinfinite cohomology of Lie-* algebras

利用紧曲线$X$上D-模范畴中的微分分次李代数胚的语言，构造了Lie-* 代数上手征模的BRST上同调复形的几何形式.

p外非分歧的pro-p伽罗瓦群的结构图之间的关系

Relationships between conjectures on the structure of pro-p Galois groups unramified outside p

我们考虑投影线负0、1和无穷大的基本群的pro-p完备化的外自同构群中有理数的绝对伽罗瓦群的规范表示。Deligne证明了由这种表示产生的分次Z_p-李代数，当用Q_p张紧时，在从3开始的每一个奇数次上都有一个元素是自由的p-进李代数，我们构造了这些元素的良好选择，并利用它们来研究Z_p-李代数的结构。特别是，我们认为它的结构如何取决于素数p的规律性通过检查格林伯格的猜想在多变量岩泽理论的后果。

Good Reduction of K3 Surfaces

Let $K$ be the field of fractions of a local Henselian DVR with perfect residue field. Assuming potential semi-stable reduction, we show that an unramified Galois-action on second $\ell$-adic cohomology of a K3 surface over $K$ implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain semi-stable flops in mixed characteristic, and study gr…

Generalized Lüroth problems, hierarchized I: SBNR -- stably birationalized unramified sheaves and lower retract rationality

This is the first of a series of papers, where we investigate hierarchies of generalized {L}üroth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal here was to construct very general necessary conditions for a smooth, not necessary proper, scheme of finite type over the perfect base field $k$ to be "retract $(-i)$-rational". We achieve this goal by constructing "stably birationalized Nisnevich subsheaf" $S_{sb}$ inside any Morel's unramified sheaf $S,$ where $S_{sb}$ coincides with $S$ on proper smooth $k$-…

K3曲面的内龙-奥格-沙法列维奇准则

A Néron-Ogg-Shafarevich criterion for K3 surfaces

本文针对K3曲面修正了内龙-奥格-沙法列维奇准则，在潜在半稳定约化假设下给出K3曲面具有好约化的等价条件，并证明了p进平展上同调的对应结果。

Cohomological dimension, Lyubeznik numbers, and connectedness properties in mixed characteristic

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.

$l$-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients

Let $(\mathsf{G},\mathsf{X})$ be a Shimura datum of Hodge type. Let $p$ be an odd prime such that $\mathsf{G}_{\mathbb{Q}_p}$ splits after a tamely ramified extension and $p\nmid |π_1(\mathsf{G}^{\rm der})|$. Under some mild additional assumptions that are satisfied if the associated Shimura variety is proper and $\mathsf{G}_{\mathbb{Q}_p}$ is either unramified or residually split, we prove the generalisation of Mantovan's formula for the $l$-adic cohomology of the associated Shimura variety. On the way we derive some new results about the geometry of the Newton stratification of the reduction…

Coxeter 型 $p$ 进 Deligne-Lusztig 概形的上同调

The cohomology of $p$-adic Deligne-Luszitg schemes of Coxeter type

本文确定了局部非阿基米德域上 Coxeter 型 $p$-Deligne-Lusztig 概形闭 Drinfeld 层的上同调，证明环面权空间集中于单一上同调度数且为成对非同构不可约表示，并探讨了 Moy-Prasad 商的极大性与 Kirillov 轨道方法的关系。

Schubert演算与李群的上同调。第一部分. 1-连通李群

Schubert calculus and cohomology of Lie groups. Part I. 1-connected Lie groups

设$G$是一个具有极大环面$T$的紧的$1$-连通李群.基于旗流形$G/T$ [15]上的Schubert演算，我们对所有$G$一致地构造了积分上同调环$H^{\ast}(G)$.

The Kottwitz conjecture for unitary PEL-type Rapoport--Zink spaces

In this paper we study the cohomology of PEL-type Rapoport-Zink spaces associated to unramified unitary similitude groups over $\Q_p$ in an odd number of variables. We extend the results of Kaletha-Minguez-Shin-White to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapport-Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

Let $G$ be a complex Lie group acting on a compact complex Hermitian manifold $M$ by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.